Explanation in Mathematical Practice

Dissertation, University of Pittsburgh (1997)

Abstract
Philosophers have paid little attention to mathematical explanations . I present a variety of examples of mathematical explanation and examine two cases in detail. I argue that mathematical explanations have important implications for the philosophy of mathematics and of science. ;The first case study compares many proofs of Pick's theorem, a simple geometrical result. Though a simple proof surfaces to establish the result, some of the proofs explain the result better than others. The second case study comes from George Polya's Mathematics and Plausible Reasoning. He gives a proof that, while entirely satisfactory in establishing its conclusion, is insufficiently explanatory. To provide a better explanation, he supplements the proof with additional exposition. ;These case studies illustrate at least two distinct explanatory virtues, and suggest there may be more. First, an explanatory improvement occurs when a sense of "arbitrariness" is reduced in the proofs. Proofs more explanatory in this way place greater restrictions on the steps that can be used to reach the conclusion. Second, explanatoriness is judged by directness of representation. More explanatory proofs allow one to ascribe geometric meaning to the terms of Pick's formula as they arise. ;I trace the lack of attention to mathematical explanations to an implicit assumption, justificationism, that only justificational aspects of mathematical reasoning are epistemically important. I propose an anti-justificationist epistemic position, the epistemic virtues view, which holds that justificational virtues, while important, are not the only ones of philosophical interest in mathematics. Indeed, explanatory benefits are rarely justificational. I show how the epistemic virtues view and the recognition of mathematical explanation can shed new light on philosophical debates. ;Mathematical explanations have consequences for philosophy of science as well. I show that mathematical explanations provide serious challenges to any theory, such as Bas van Fraassen's, that considers explanations to be fundamentally answers to why-questions. I urge a closer interaction between philosophy of mathematics and philosophy of science; both will be needed for a fuller understanding of mathematical explanation.
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Mathematical Explanation and the Theory of Why-Questions.David Sandborg - 1998 - British Journal for the Philosophy of Science 49 (4):603-624.
Against Mathematical Explanation.Mark Zelcer - 2013 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 44 (1):173-192.

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Mathematical Explanation and the Theory of Why-Questions.David Sandborg - 1998 - British Journal for the Philosophy of Science 49 (4):603-624.
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